A metacontinuum model for phase gradient metasurfaces

Acoustic metamaterials and metasurfaces often present complex geometries and microstructures. The development of models of reduced complexity is fundamental to alleviate the computational cost of their analysis and derivation of optimal designs. The main objective of this paper is the derivation and validation of a metacontinuum model for phase gradient-based metasurfaces. The method is based on the transformation acoustics framework and defines the metasurface in terms of anisotropic inertia and bulk modulus. Thermal and viscous dissipation effects in the metacontinuum are accounted for by introducing a complex-valued speed of sound. The model is implemented in a commercial FEM code, and its predictions are compared with numerical simulations on the original geometry and also using an equivalent boundary impedance approach. The results are examined for an exterior acoustics benchmark and for an in-duct installation in terms of transmission coefficient with the four-pole matrix method. The metacontinuum model gives solid results for the prediction of the acoustic properties of the examined metasurface samples for all the analyzed configurations, as accurate as the equivalent impedance model on which it is based and outperforming it in some circumstances.


A metafluid model for phase gradient metasurfaces Acoustic impedance model for a Helmholtz resonator
In the main article, an impedance model for the Helmholtz resonator cells is used. We report here some detail on its derivation. In the equation of motion of the mass-spring system, representing the resonator in Fig.S1, the displacement of the mass element y n can be written as a function of the pressure perturbation on the neck section Figure S1. The design of the Helmholtz resonator defines its surface impedance, from which an equivalent length, function of frequency, for a straight duct with the same impedance can be derived. The properties of the metafluid are then evaluated, arbitrarily setting its domain thickness, using the procedure described in the article. m d 2 y n dt 2 = (p + ∆p)r n , m = ρ 0 r nĥn (S1) Within the limit of small variations for the involved variables, we can write where d p dρ = c 0 and dv = r n y n ρ 0 V , with V = h c r c . Hence, substituting in the previous, the governing equation for the displacement y n becomes m d 2 y n dt 2 + Considering harmonic variations of the quantities p =p n e iωt y =ỹ n e iωt , Eq.S3, and dropping the time exponential terms, one obtains − ω 2 mỹ n + c 2 0 ρ 0 r 2 n Vỹ n =pr n (S4) Recalling that the acoustic particle velocity isũ = iωỹ n , the expression for the specific acoustic impedance is retrieved The termĥ n is the length of the neck of the resonator, accounting for the end corrections as described by Bies & Calton for each side of the neck 1, 2 h n = h n + 2 0.85r n 1 − 1.33 r n r c (S6)

Design variables for PGMS cells
The HR and SC cells used have been designed through the optimization procedure described in 3 . Here, in Tab. S1 and Tab. S2, we report the values obtained for a set of eight cells for each type.

Space-coiling cells
We report in Figs. from S2 to S8 some additional results for the space-coiling cells, in terms of transmission coefficient.
(b) SC cell #1 with losses. Figure S2. SC cells #1, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) SC cell #2 with losses. Figure S3. SC cells #2, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) SC cell #3 with losses. Figure S4. SC cells #3, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k). Figure S5. SC cells #4, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) SC cell #5 with losses. Figure S6. SC cells #5, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) SC cell #6 with losses. Figure S7. SC cells #6, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) SC cell #7 with losses. Figure S8. SC cells #7, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).

Helmholtz resonator cells
In Figs. from S9 to S14, we report the results in terms of transmission loss for some of the Helmholtz Resonator cells 6/9 (a) HR cell #3 without losses.
(b) HR cell #3 with losses. Figure S9. HR cells #3, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) HR cell #4 with losses. Figure S10. HR cells #4, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) HR cell #5 with losses. Figure S11. HR cells #5, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) HR cell #6 with losses. Figure S12. HR cells #6, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k). Figure S13. HR cells #7, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).
(b) HR cell #8 with losses. Figure S14. HR cells #8, comparison of transmission coefficient spectra between full geometry, equivalent impedance, and equivalent metafluid lossless simulations (a) and including thermoviscous losses (b). For the lossy case , narrow regions approximation (NR approx) and a full thermoviscous acoustic simulation (TV acoustics) are compared with the equivalent impedance and metafluid modelling with complex wavenumber (complex k).